Generalized bent functions with large dimension (Q6584335)

Fix integers \(n,k>0\), a prime \(p\), and let \(\mathrm{GF}(p)\) denote the finite field with \(p\) elements (identified, when convenient, with \(\{0,1,\dots,p-1\}\)). Recall a bent function \(f: \mathrm{GF}(p)^n\to \mathrm{GF}(p)\) is a function which is, roughly speaking, as far away as possible from being linear (in the sense that the absolute value of the Hadamard-Fourier transform of \(f\) is a constant and as large as possible). Previously, Mesnager and others characterized generalized bent functions \(f: \mathrm{GF}(p)^n\to {\mathbb{Z}}/p^k{\mathbb{Z}}\) in terms of a bent function \(g: \mathrm{GF}(p)^n\to \mathrm{GF}(p)\) and a partition \(P=P_g\) of \(\mathrm{GF}(p)^n\) satisfying certain conditions.\N\NMesnager's characterization is the starting point of this article under review. Meidl's clever idea is, roughly speaking, to weaken the characterization of Mesnager et al, study the dimension of an affine space attached to this weaker notion, then re-characterize generalized bent functions using this dimension.\N\NThe basic idea of the way it works is as follows: given \(f\) and \(g\) as above, there are functions \(a_i: \mathrm{GF}(p)^n\to \mathrm{GF}(p)\) such that\N\[\Nf(x) = a_0(x)+a_1(x)p+\dots+ a_{k-2}(x)p^{k-2}+g(x)p^{k-1}.\N\]\NNote these \(a_i\) and \(g\) are uniquely determined from \(f\). Consider the affine space\N\[\NA = A_f = \{g(x) + F(a_0(x), \dots, a_{k-2}(x))\mid F: \mathrm{GF}(p)^{k-1}\to \mathrm{GF}(p)\}.\N\]\NThe author observes that the dimension of \(A\) is the cardinality of the associated partition \(P\). In this context, the author studies questions such as:\N\begin{itemize}\N\item[(1)] What dimensions arise in this way?\N\item[(2)] What partitions arise in this way?\N\end{itemize}\NThe author analyzes these sorts of questions for various classes of bent functions. For example, the author shows that Maiorana-McFarland bent functions allow for the largest possible partitions, resulting in affine bent function spaces of dimension \(p^{n/2}\).\N\NA few of the main theorems (that are also relatively simple to state) in this paper are:\N\begin{itemize}\N\item Theorem 3.2, which provides a partition for the Maiorana-McFarland bent function \(f: \mathrm{GF}(p)^n\times \mathrm{GF}(p)^2\to \mathrm{GF}(p)\), \(f(x,y,z)=f_y(x)-yz\), determined from the partitions for \(f_0, \dots, f_{p-1}\).\N\item Theorem 3.4, which states that the generalized Maiorana-McFarland construction yields a generalized bent function over \(\mathrm{GF}(p)^{n+2}\) of maximal possible dimension \(p^{(n+2)/2}\) (for the affine space \(A\) mentioned above).\N\end{itemize}\NIn addition, the author investigates partitions attached to various types of bent functions, including weakly regular, non-weakly regular, and non-dual bent functions. The author also constructs partitions for bent functions derived from bent partitions, such as the partial spread class, and partitions for Carlet's classes C and D. Unfortunately, these results are significantly more technical to state. For more precise details of these results, the reader is referred to this interesting and important paper itself.